Questions tagged [sheaf-cohomology]

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Cohomology of equivariant toric vector bundles using Klyachko's description

I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties. Whereas detailed literature ...
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exact functor in syntomic cohomology

By Tag. 04C4 of the Stacks Project, for $f:X\rightarrow Y$ a closed immersion of schemes, the pushforward $f_*$ is exact for abelian sheaves on the big syntomic site. Is it also true for a finite ...
prochet's user avatar
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Reference for isomorphism between group cohomology and singular cohomology

Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that $$ H^i(G, ...
Aidan's user avatar
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Action of algebraic group in cohomology of equivariant algebraic vector bundle

Let $X$ be a projective algebraic variety over an algebraically closed field. Let an algebraic group $G$ act algebraically on $X$. Let $\mathcal{F}$ be a $G$-equivariant vector bundle (or, more ...
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A question about cohomology with local coefficient

Let's consider the next theorem. Theorem [The cohomology Leray-Serre Spectral sequence] Let $R$ be a commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{% \rightarrow }B$, ...
Mehmet Onat's user avatar
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1 answer
252 views

Commutative group scheme cohomology on generic point

Setup: Let $k$ be an algebraically closed field. Let $C$ be a smooth connected projective curve over $k$. Let $J$ be a smooth commutative group scheme over $C$ with connected fibers. Let $j:\eta\to C$ ...
lzww's user avatar
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Computing the first sheaf cohomology

I am looking for some examples of computing the dimension of the first sheaf cohomology for smooth projective surfaces. To be more precisely, let $X$ be a smooth, projective surface. Let $D$ be an ...
Leo D's user avatar
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In what sense is the complex $\mathscr{L}^\bullet$ unique?

This is in Section III.12. Algebraic Geometry by Hartshorne. Assume $X\to\mathrm{Spec}(A)$ is a projective morphism of Noetherian schemes. Let $\mathscr{F}$ be coherent over $X$, flat over $A$. ...
Display Name's user avatar
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1 answer
244 views

If $\mathrm{Ext}^i(E,F)$ commutes with base change, then is $\mathrm{Ext}^{i+1}(E,F)$ representable?

Consider a projective morphism of Noetherian schemes $p:X\to \mathrm{Spec}(A)$. Let $\mathcal{E},\mathcal{F}$ be coherent $\mathcal{O}_X$-modules flat over $A$. For every (Noetherian) ring map $A\to B$...
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Clarification on smooth de Rham theorem

I am misunderstanding something in Theorem 2.1.9 in Dimca’s Sheaves in Topology: Let $X$ be a real smooth manifold. Then the natural morphism from the constant sheaf to the de Rham complex $$\mathbb{R}...
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How to estimate the locus of non-zero cohomology for a equivariant toric reflexive sheaf, with a Klyachko description

I am trying to analyze Macaulay2 package "ToricVectorBundles". The package deals with equivariant reflexive sheaves on complete toric varieties. Such a sheaf is described by a set of ...
Maciej Gałązka's user avatar
2 votes
3 answers
457 views

Canonical product in sheaf cohomology

EDIT: Let $\mathcal{F},\mathcal{G}$ be sheaves of abelian groups on a topological space $X$. Then there exists a canonical cup product $$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...
asv's user avatar
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On the exactness of the restriction to small etale site in the case of differentiable stacks

Let $X$ be a manifold considered as a stack on the big etale site of manifolds. Consider the functor $$ F \mapsto F_X$$ which sends a sheaf on the stack $X$ to the sheaf on the manifold $X$. In Lemma ...
Spai's user avatar
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When (or why) is a six-functor formalism enough?

The six functor formalism in a given cohomology theory consists of for each space a derived category of sheaves and six different ways to construct functors between those categories (four involving a ...
Will Sawin's user avatar
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Čech-like cohomology with the “other nerve”

Let $X$ be a space and $\mathcal U$ a cover of $X$. Instead of Čech cohomology, I would like to take the following construction: let $$I= \{ \text{finite nonempty intersections of elements of }\,\...
Joshua Mundinger's user avatar
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Exterior product of Euler Exact Sequence

Consider the Euler exact sequence: $ 0\longrightarrow \mathcal{O}_{\mathbb{P}^n} \longrightarrow \mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\longrightarrow \mathcal{T}_{\mathbb{P}^n} \longrightarrow 0 $ This ...
BVquantization's user avatar
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Sheaf cohomology definition

I have seen multiple definitions for sheaf cohomology and wanted to ask for the reason. One goes through injective resolutions and the other through flasque resolutions. For paracompact bases it holds ...
katle's user avatar
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Why does the associated sheaf vanish?

I am learning local cohomology from Hartshorne’s book Local Cohomology. My question is about understanding a line in the proof of proposition 1.11 in this book. The set-up for proposition 1.11 is that ...
Boris's user avatar
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Sheaf cohomology in number theory

I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...
Tuvasbien's user avatar
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Using higher topos theory to study Cech cohomology

It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...
Markus Zetto's user avatar
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189 views

Singular cohomology to cohomology of quasi-coherent sheaf

Let $X$ be a projective nonsingular variety (integral Noetherian scheme of finite type that is proper over a field $k=\overline{k}$ such that $\Omega^1_X$ is locally free). Suppose one knows the ...
locally trivial's user avatar
6 votes
1 answer
440 views

Unbounded acyclic resolutions

Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...
R. van Dobben de Bruyn's user avatar
11 votes
1 answer
362 views

Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper

In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
algori's user avatar
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Long exact sequence in Borel-Moore homology

The Wikipedia page for Borel-Moore homology states that for a locally compact set $X$ and a closed subset $Z$, if we write $U = X \setminus Z$ we have the following long exact sequence $$\cdots \to H^{...
Eduardo de Lorenzo's user avatar
2 votes
0 answers
108 views

Homotopy invariant Bloch-Ogus cohomologies with a vanishing property

I am looking for examples (in any characteristic) of homotopy invariant Bloch-Ogus cohomology theories given by Zariski sheaves $\Gamma(n)$, such that $\Gamma(0) = \mathbb{Z}$ is the constant sheaf. ...
user127776's user avatar
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2 votes
1 answer
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Surjection of a short exact sequence induced by spectral sequence (from paper of Schneider/Stuhler)

Let $K=\mathbb{Q}_p$ and $X$ a smooth separated rigid analytic variety over $K$ with coherent sheaf $\mathcal{F}$. Furthermore, $U \subset X$ is an open subvariety with admissible covering $$ \dotsb \...
KKD's user avatar
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0 answers
193 views

Is $\mathbb{C}^*$ not irreducible, or is every locally constant sheaf on $\mathbb{C}^*$ constant?

I am running into contradiction from the following set of definitions, propositions, and assumptions. Can anyone spot where I'm off? Definition A sheaf $\mathcal{F}$ on a topological space $X$ is ...
locally trivial's user avatar
2 votes
1 answer
217 views

Formula for the Euler characteristic of a local system on $\mathbb{P}^1$

Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion. Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...
stupid_question_bot's user avatar
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0 answers
58 views

Isomorphism of cohomology bundles for smooth homotopic fibrations

Let $M,N$ be closed smooth manifolds. Let $f_0,f_1\colon M\to N$ be two smooth fibrations which are homotopic to each other in the class of smooth (equivalently, continuous) maps. Let $E^i_0, E^i_1$ ...
asv's user avatar
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16 votes
1 answer
429 views

Zorn's lemma for Grothendieck sites

In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction ...
cat man's user avatar
  • 163
6 votes
0 answers
208 views

Is the right adjoint to presheaf direct image interesting?

Let $X\overset{f}{\to}Y$ be a continuous map. It induces on presheaves a classical adjunction inverse image ⊣ direct image. However, the direct image functor has a further right adjoint, defined by ...
Arrow's user avatar
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6 votes
1 answer
422 views

Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?

Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
Jakob Werner's user avatar
  • 1,083
2 votes
1 answer
176 views

On the upper-bound for a type of quintuple Kloosterman sums

Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound. My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...
hofnumber's user avatar
  • 553
3 votes
1 answer
331 views

Estimates for certain double-Kloosterman sums

Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here. For any $q\in \mathbb{N}^+$, how can we estimate the type of sum $$ \sideset{_{}^{}}{^{\ast}_{}}\...
hofnumber's user avatar
  • 553
1 vote
1 answer
219 views

A question involving the three-dimensional Kloosterman sum

Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here. For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$,...
hofnumber's user avatar
  • 553
5 votes
0 answers
228 views

Cohomology of coherent sheaves on Deligne Mumford stacks

Suppose that $\cal X$ is tame Deligne Mumford stack with generic trivial inertia. Let $X$ be its muduli space and $f:{\cal X}\to X$ the projection. Let $\cal F$ be a coherent sheaf on $\cal X$. Is it ...
Carletto's user avatar
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5 votes
0 answers
248 views

What's the point of fine sheaves? (As opposed to soft ones)

Why do people care to define fine sheaves? What useful property do they have for which softness is not sufficient? some observations (because I feel guilty about a the one-line question): The point ...
Carlos Esparza's user avatar
3 votes
0 answers
168 views

Deligne's integrality theorem in the setting of $ \mathbb{F}_{\ell}((t)) $-adic cohomology

Let $ \mathbb{F}_{q} $ be a finite field of characteristic $ p $ and $ \overline{\mathbb{F}_{q}} $ be an algebraic closure of $ \mathbb{F}_{q} $. Let $ X $ be a smooth projective variety over $ \...
Nobody's user avatar
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1 vote
0 answers
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Artin-Winters proof of semi-stable reduction theorem: details

I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail— Let $\...
BelowAverageIntelligence's user avatar
3 votes
0 answers
175 views

Hypercovers with sieves

Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\...
Leo Herr's user avatar
  • 1,004
3 votes
0 answers
117 views

Is the cohomology $H^1(X, \mathcal{E}^\nabla)$ trivial, for the sheaf of constants of an algebraic connection $\nabla$?

Suppose that $\pi:X\to S$ is a flat morphism between Noetherian, integral schemes (of characteristic zero, if need be). Let $\mathcal{E}$ be a locally free sheaf on $X$, and $$\nabla:\mathcal{E} \to \...
PrimeRibeyeDeal's user avatar
1 vote
0 answers
308 views

Global section of pullback of an ideal sheaf

For a local ring $R$ with maximal ideal $\mathfrak{m}\subset R$ and residue field $\kappa$, and a flat morphism $f\colon X\rightarrow \mathrm{Spec} R$ of schemes, we consider the short exact sequence ...
Takagi Benseki's user avatar
2 votes
0 answers
51 views

Conditions for long exact sequence for line bundles on curve to degenerate?

Let $\varphi:X\to Y$ be a morphism of schemes of relative dimension 1, and $\mathcal{L}' \xrightarrow{g} \mathcal{L}$ an injection of line bundles on $X$. The sequence $$0\to \mathcal{L}' \xrightarrow{...
PrimeRibeyeDeal's user avatar
4 votes
1 answer
286 views

Type vs degree of a polarized abelian variety

Let $(A,L)$ be a polarized abelian variety. I know that the degree of the polarization is the Euler characteristic of $L$, so that $d = \chi(L) = \dim H^0(A,L)$ since $L$ is ample. I've read in a lot ...
TartagliaTriangle's user avatar
1 vote
0 answers
94 views

Cohomology with coefficient in sheaf of morphisms of an algebraic group

Let $G$ be an affine algebraic group over ${\mathbb C}$. We denote the sheaf of morphisms from ${\mathbb A}^1$ to $G$ by $\bf G$. Then $H^1({\mathbb A}^1,\bf G)=0$ (Cech cohomlogy). Is this fact true? ...
piper1967's user avatar
  • 1,049
2 votes
2 answers
262 views

Extensions for a short exact sequence on Grassmannians

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Ext{Ext}$Let us consider a $n$-dimensional complex vector space $V$ and denote by $G(k,n)$ the Grassmannian of $k$-planes in $V$. We use the ...
Bobech's user avatar
  • 381
1 vote
0 answers
120 views

Base change of cohomology when the cohomology is a torsion

Let $(R,m,k)$ be a discrete valuation ring, where $k=R/m$. Let $X\rightarrow \mathrm{Spec}R$ be a projective, integral and flat $R$-scheme. Let $\mathscr F$ be a coherent sheaf such that $H^i(X,\...
Pickle Liobe's user avatar
1 vote
2 answers
354 views

Pushforward of structure sheaf along a torsor for a finite group

Let $\pi : P \to X$ be a torsor for a discrete, finite group $G$ of size $\#G = N$ on a scheme $X$. I want to compare $\pi_* \mathcal O_P$ with $\mathcal{O}_X$. Locally but not globally, $\pi_* \...
Leo Herr's user avatar
  • 1,004
5 votes
1 answer
268 views

First cohomology of tangent sheaf of rational curve

Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$. Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of ...
Jef's user avatar
  • 909
6 votes
0 answers
146 views

Cech cohomology on product covers & Fréchet sheaves

My question is about the paper [Ka67]. Let $S, T$ be sheaves of nuclear Fréchet spaces over paracompact topological spaces $X, Y$, respectively; in particular, if $V \subset U$ are open subsets in $X$,...
Lukas Miaskiwskyi's user avatar

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